Schäffer spaces and uniform exponential stability of linear skew-product semiflows

J. Differential Equations 212 (2005) 191–207

www.elsevier.com/locate/jde

Schäffer spaces and uniform exponential stability oflinear skew-product semiflowsPetre Preda∗, Alin Pogan, Ciprian Preda

Department of Mathematics, West University of Timi¸soara, Bd. V. Pârvan, No 4,Timisoara 1900, Romania

Received 1 March 2004; revised 26 July 2004

Available online 9 September 2004

Abstract

We study the exponential stability of linear skew-product semiflows on locally compactmetric space with Banach fibers. Our main tool is the admissibility of a pair of the so-calledSchäffer spaces. This characterization is a very general one, it includes as particular cases manyinteresting situations among them we can mention some results due to Clark, Datko, Latushkin,van Minh, Montgomery–Smith, Randolph, Räbiger, Schnaubelt.© 2004 Elsevier Inc. All rights reserved.

MSC: primary 34D05; 35B40; 93D05

Keywords:Linear skew-product semiflow; Schäffer spaces; Uniform exponential stability; Admissibility

1. Introduction

A classical and well-studied subject in the field of differential equations is the theoryof linear skew-product (semi)flows, which arise as solution operators for variationalequation

d

dtu(t) = A(�(�, t))u(t),

∗ Corresponding author.E-mail addresses:[email protected](P. Preda),[email protected](C. Preda).

0022-0396/$ - see front matter © 2004 Elsevier Inc. All rights reserved.doi:10.1016/j.jde.2004.07.019

http://www.elsevier.com/locate/jde

mailto:[email protected]

mailto:[email protected]

192 P. Preda et al. / J. Differential Equations 212 (2005) 191–207

where � is a semiflow on a locally compact spaces andA(�) an unbounded linearoperator on a Banach spaceX, for each� ∈ �.Also, the qualitative behavior of (semi-)flows on (locally) compact spaces or

(�-) finite measure spaces is described by notions like stability or exponential di-chotomy of the associated linear skew-product (semi-)flow. In the finite-dimensionalcase the Sacker–Sell spectrum permits an important and useful characterization of theseproperties (see[9,20–22]), which was extended recently to norm-continuous cocycles oninfinite-dimensional Banach spaces by Latushkin and Stepin. However, all truly infinite-dimensional situations, e.g. flows originating from partial differential equations andfunctional differential equations, only yield strongly continuous cocycles. This causesserious problems and new phenomena. In recent decades, significant progress has beenmade in the study of asymptotic behavior of linear skew-product flows and nonau-tonomous Cauchy problems (see[7,10–12,16,19,20]and the literature cited therein),giving an unifield answer to an impressive list of classical problems. Also, in last fewyears important contributions were done in infinite-dimensional case (see[2,3,5,22])and in the applications. There has been studied the dichotomy of linear skew-productsemiflows defined on compact spaces (see[2–5]), and on a locally compact spaces,respectively (see[15]). An answer concerning the exponential stability and exponentialdichotomy of linear skew-product flow on locally compact metric space� has beendone by the investigation begun in[13,14] using the so-called evolution semigroup.In [15], dichotomy of strongly continuous linear skew-product semiflows was ex-

pressed in terms of hyperbolicity of a family of weighted shift operators and thus itwas extended the classical theorem of Perron[18].Arguments in these papers again illustrate the general philosophy of “autonomization’’

of nonautonomous problems by passing to associated evolution semigroups.In contrast to this “philosophy’’, the present paper shows that we can characterize

the exponential stability in terms of the admissibility of some suitable pairs of spacesin a direct way, without so-called evolution semigroups. But, this is not the first aimof our paper.Our attention is devoted to the relationship between uniform exponential stability of a

linear skew-product semiflow on a locally compact metric space and the admissibilityof a pairs of spaces which are translation invariant, the so-called Schäffer spaces.Therefore, it is considered a concept of exponential stability for linear skew-productsemiflows, which is an extension of the classical concept of exponential stability fortime-dependent linear differential equations in Banach spaces (see e.g.[6–8,16,17]). Intheir monumental works of the 1960s, Massera and Schäffer present an indepth study ofexponential dichotomies for nonautonomous linear equations in Banach spaces. A keyconcept used in their study is the notion ofadmissible spaces. Roughly speaking, thesespaces represented the input and output spaces for a linear inhomogeneous problem.From this point of view our work fit into Massera–Schäffer context. Until now, themost common classes of spaces used for the connection between admissibility andstability areLp or Mp spaces, which are in particular Schäffer spaces, so with thisapproach it can be obtained an unified treatment, generalizing the above results. Also,as we already note, many analysis concerning the relation between the admissibilityand exponential stability involves the evolution semigroup, so the “input space’’ and

P. Preda et al. / J. Differential Equations 212 (2005) 191–207 193

the “output space’’ must be the same. Here, the proofs are directly, using the methodsof the so-called “test’’ functions, so we can extend the analysis to the case when the“input space’’ is different from the “output space’’ and the requirement above can bedropped. Moreover, we remark here that even the Orlicz spaces are Schäffer spacestoo, and this approach does break a new ground, bringing into attention other usefulsituations and adding some nice twist to the subject concerning the connection betweenexponential stability and admissibility.

2. Preliminaries

In the beginning let us fix some standard notations. ForX a Banach space we denoteby M(R+, X) the space of all Bochner measurable functions fromR+ to X and by:

L1loc(R+, X) ={f ∈ M(R+, X) :

∫K

||f (t)|| dt < ∞, for each compactK in R+},

Lp(R+, X) ={f ∈ M(R+, X) :

∫R+

||f (t)||p dt < ∞}, wherep ∈ [1,∞),

L∞(R+, X) = {f ∈ M(R+, X) : ess supt∈R+

||f (t)|| < ∞},

Mp(R+, X) ={f ∈ M(R+, X) : sup

t∈R+

∫ t+1

t

||f (s)||p ds < ∞}, wherep ∈ [1,∞).

T (R+, X) the space of all functionsf ∈ L1loc(R+, X) with the property that thereexist (�n)n∈N and (an)n∈N two sequences of positive real numbers such that

∞∑n=0

an < ∞ and ||f ||�∞∑n=0

an�[�n,�n+1].

Also for � a locally compact metric space we denote byCb(�, X) the space of allbounded continuous functions from� to X.We note thatLp(R+, X), L∞(R+, X), Mp(R+, X), Cb(�, X) are Banach spaces

endowed with the, respectively, norms:

||f ||p =(∫

R+||f (t)||p dt

) 1p

,

||f ||∞ = ess supt∈R+

||f (t)||,

||f ||Mp = supt∈R+

(∫ t+1

t

||f (s)||p ds) 1p


and

||f ||T = inf

{ ∞∑n=0

an : where(an)n∈N satisfies the above inequality

}

||f ||Cb(�,X) = sup�∈�

||f (�)||.

In order to simplify the notations we putLp := Lp(R+,R), L∞ := L∞(R+,R),Mp := Mp(R+,R), for all p ∈ [1,∞) and L1loc = L1loc(R+,R), T = T (R+,R),||f ||Cb(�,X) = |||f |||.Next, we remind the definition of Schäffer spaces.

Definition 2.1. A Banach spaceE is said to be a Schäffer space if the followingproperties are satisfied:(s1) E ⊂ L1loc(R+,R) and for any compactK ⊂ R+ there exists�K > 0 such that

∫K

|f (t)| dt��K ||f ||E for all f ∈ E

(s2) �[0,t] ∈ E, for all t�0, where�[0,t] denotes the characteristic function of theinterval [0, t]

(s3) If f ∈ E and h ∈ M(R+,R) with |h|� |f |, thenh ∈ E and ||h||E� ||f ||E .(s4) If f ∈ E, t�0, gt : R+ → R, gt (s) =

{0, s ∈ [0, t),f (s − t), s ∈ [t,∞),

thengt ∈ E and

||gt ||E = ||f ||E .

Now, we present some good examples which can be used to describe the importanceof this approach.

Example 2.1. It is easy to check thatMp, Lp,Lp∩Lq , L∞ andT, the spaces mentionedabove, are Schäffer spaces. One can easy remark thatT ⊂ E ⊂ M1, for any SchäfferspaceE (for more results in this direction see[16]).Another remarkable example of Schäffer spaces are the Orlicz spaces. Let� : R+ →

R+ be a function which is nondecreasing, left-continuous with�(t) > 0, for all t > 0.Define

�(t) =∫ t

0�(s) ds.


A function � of this form is called a Young function. Forf : R+ → R a measurablefunction and� a Young function we define

M�(f ) =∫ ∞

0�(|f (s)|) ds.

The setL� of all f for which there exists ak > 0 thatM�(kf ) < ∞ is easilychecked to be a linear space. With the norm

�(f ) = inf

{k > 0 : M�

(1

kf

)�1

}

the space(L�,�) becomes a Banach space which is easy to see that verify theconditions (s2)–(s4). In order to check the condition(s1) considerf ∈ L�, t > 0,k > 0, such thatM�(1

kf )�1. Then we have that

�(1

kt

∫ t

0|f (s)| ds

)� 1

t

∫ t

0�

(1

k|f (s)|

)ds� 1

t

and so

∫ t

0|f (s)| ds� t�−1

(1

t

)k,

which implies that

∫ t

0|f (s)| ds� t�−1

(1

t

)�(f )

for all f ∈ L�, t > 0, and hence the condition(s1) is also verified, so the Orliczspaces are Schäffer spaces too.From here we can obtain other interesting examples. In order to find smaller function

spaces which can be used as “input spaces’’, in our approach, we present below afunction space which is contained in allLp, for p ∈ [1,∞). For instance, if we take�(t) = et − 1, thenL� ⊂ Lp, for all p ∈ [1,∞).Indeed one can see thattm�m!�(t) for all t�0 and allm ∈ N∗ which implies

that L� ⊂ Lm, for all m ∈ N∗. Having in mind thatLm⋂Lm+1 ⊂ Lp for all

p ∈ [m,m+ 1], and allm ∈ N∗, it follows that L� ⊂ Lp, for all p ∈ [m,m+ 1], andall m ∈ N∗.

If E is a Schäffer space we denote by

E(X) = {f ∈ M(R+, X) : t �→ ||f (t)|| : R+ → R is in E}.


Remark 2.1. E(X) is a Banach space endowed with the norm

||f ||E(X) = || ||f (·)|| ||E.

Proof. First, let us prove that ifu ∈ E then |u| < ∞ a.e. Indeed considerA = {t ∈R+ : |u(t)| = ∞}. It is easy to check that

|u|� |u|�A��A for all > 0

and so by(s3), one can easily see that

�A ∈ E and‖�A‖E = 0.

Now, it is clear thatm(A) = 0 and so|u| < ∞ a.e. In order to prove thatE(X) is aBanach space, let{fn}n∈N be a Cauchy sequence inE(X). We will prove that thereexists {fnk }k∈N a subsequence of{fn} and f ∈ E such that

fnkE(X)−→ f, fnk → f a.e.

From the fact that{fn}n∈N is Cauchy sequence then there exists{fnk }k∈N a subse-quence of{fn} such that

||fnk+1 − fnk ||E(X)�1

2kfor all k ∈ N.

It follows that the sum(∑

k�0 ||fnk+1 − fnk ||)is absolute convergent in the Banach

spaceE. Let gE= ∑∞

k=0 ||fnk+1 − fnk ||, gm = ∑mk=0 ||fnk+1 − fnk || and h : R+ →

R+, h(t) = ∑∞k=0 ||fnk+1(t)− fnk (t)||. It is easy to see that

(∗) gmE−→ g and gm(t) ↗ h(t) for all t�0.

On the other hand,

∫ t

0|g(s)− h(s)| ds �

∫ t

0|g(s)− gm(s)| ds +

∫ t

0|gm(s)− h(s)| ds

� �E(t)||gm − g||E +∫ t

0h(s) ds −

∫ t

0gm(s) ds

for all t�0 and allm ∈ N.


Using (∗) we obtain that

∫ t

0|g(s)− h(s)| ds = 0 for all t�0

and henceg = h a.e.Let B = {t ∈ R+ : |g(t)| = ∞}, C = {t ∈ R+ : g(t) �= h(t)}, D = B

⋃C. Using

the fact thatg ∈ E and g = h a.e., we have that

m(D) = 0 and∞∑k=0

||fnk+1(t)− fnk (t)|| < ∞ for all t ∈ R+ \D,

which implies that the subsequence{fnk (t)}k∈N is convergent inX for all t ∈ R+ \D.Let f : R+ → X the map given by,

f (t) ={

limk→∞ fnk (t), t ∈ R+ \D,0, t ∈ D.

Then we have thatfnk → f a.e. and sof is measurable and

f (t)− fnm(t) =∞∑k=m

(fnk+1(t)− fnk (t))

for all t ∈ R+ \D and allm ∈ N and hence

||f (t)− fnm(t)||�g(t)− gm−1(t) for all t ∈ R+ \D and all m ∈ N∗.

It follows that f − fnm ∈ E for all m ∈ N∗ and sof ∈ E and

||f − fnm ||E(X)� ||g − gm−1||E for all m ∈ N∗.

We obtain thatfnkE(X)−→ f, fnk → f a.e. Because{fn}n∈N is a Cauchy sequence

with a convergent subsequence it follows that{fn}n∈N is convergent. �

Remark 2.2. If {fn}n∈N ⊂ E(X), f ∈ E(X), fn → f in E(X), then there exists{fnk }k∈N a subsequence of{fn}n∈N such that

fnk → f a.e.


Proof. If fnE(X)−→ f then {fn}n∈N is a Cauchy sequence and hence from what we have

already mentioned in the proof of the Remark 2.1 there exists{fnk }k∈N a subsequenceof {fn} and g ∈ E such that

fnkE(X)−→ g, fnk → g a.e.

This implies thatf = g a.e. and sofnk → f a.e.For a Schäffer spaceE we denote by�E,�E : R+ → R+ the following applications:

�E(t) = inf

{� > 0 :

∫ t

0|f (s)| ds��||f ||E for all (t, f ) ∈ R+ × E

},

�E(t) = ||�[0,t]||E.

It is known (see[16]) that �E,�E are nondecreasing functions and moreover

(∗∗) t��E(t)�E(t)�2t for all t�0. �

Example 2.2. It is easy to see that forLp andMp we have:

�Lp(t) ={t1− 1

p , p ∈ [1,∞), t�0,t, p = ∞, t�0,

�Lp(t) ={t1p , p ∈ [1,∞), t�0,1, p = ∞, t�0,

t��Mp(t)�[t] + {t}1− 1p , for each(p, t) ∈ [1,∞)× R+, where[t]

denotes the largest integer less or equal thant and{t} = t − [t].

�Mp(t) ={t1p , t ∈ [0,1),1, t�1,

�L�(t) = t�−1(1t),

�L�(t) =(�−1(1

t))−1

.


Definition 2.2. A mapping� : � × R+ → � is called a semiflow on� if it has thefollowing properties:(sf1) �(�,0) = � for all � ∈ �;(sf2) �(�, t + s) = �(�(�, s), t) for all t, s�0, � ∈ �;(sf3) � is continuous.

In order to simplify the notations we will put�(�, t) = �t

Definition 2.3. A pair � = (�,�) is called linear strongly continuous skew-productsemiflow onE = X × � if � is a semiflow on� and� : � × R+ → B(X) satisfythe following conditions:(sp1) �(�,0) = I (the identity operator onX);(sp2) �(�, t + s) = �(�(�, t), s)�(�, t) for all t, s�0, � ∈ �;(sp3) � is strongly continuous;(sp4) there existsM, > 0 that

||�(�, t)||�Me t for all t�0.

Example 2.3. Let X be a Banach space,� a locally compact metric space,T ={T (t)}t�0 a C0-semigroup and{U(�)}�∈� a bounded strongly continuous family ofidempotent operators with the property that

U(�)T (t) = T (t)U(�) for all t�0, � ∈ �,

then the pair� = (�,�) defined by

�(�, t) = �,�(�, t) = U(�)T (t)

is a strongly continuous skew-products semiflow.

Example 2.4. Let � be a compact metric space,� a semi-flow on�, X a BanachspaceA : � → B(X) a continuous mapping. If�(�, t)x is the solution of the abstractCauchy problem

{u′(t) = A(�(�, t))u(t), t�0,u(0) = x,

then the pair� = (�,�) is a strongly continuous skew-products semi-flow.

Also, a large numbers of examples of strongly continuous skew-products semi-floware provided in the recent literature (see for instance[14]).


Definition 2.4. A strongly continuous skew-products semiflow� = (�,�) is calleduniformly exponentially stable (u.e.s) if there existsN, � > 0 such that

||�(�, t)||�Ne−�t for all t�0, � ∈ �.

In what follows forf ∈ L1(R+, X) we definexf : � × R+ → X the map given by

xf (�, t) =∫ t

0�(�s, t − s)f (s) ds.

From the Definition 2.3 it is easy to see thatxf (·, t) ∈ Cb(�, X) for all t�0.

Definition 2.5. The pair (E, F ) is said to be admissible to� if for all f ∈ E(X) themap uf : R+ → Cb(�, X) given by uf (t) = xf (·, t) belongs toF(Cb(�, X)).

If we are in the case of strongly continuous skew-products semiflow provided byExample 2.4 then for all� ∈ �, xf (�, ·) is the solution of the Cauchy problem

{u′(t) = A(�(�, t))u(t)+ f (t), t�0,u(0) = 0.

3. The main results

Let (E, F ) be a pair of Schäffer spaces

Lemma 3.1. If the pair (E, F ) is admissible to�, then there isK > 0 such that

||uf ||F(Cb(�,X))�K||f ||E(X)

Proof. We set nowV : E(X) → F(Cb(�, X)), Vf = uf . It is obvious thatV is alinear operator.If we consider {fn}n∈N ⊂ E(X), f ∈ E(X),g ∈ F(Cb(�, X)) such that

fnE(X)−→ f, Vfn

F(Cb(�,X))−→ g

then, by Remark 2.2, there exists a subsequence{fnk }k∈N of {fn}n∈N such that

fnk → f a.e. for k → ∞, VfnkF (Cb(�,X))−→ g a.e. for k → ∞


||xfnk (�, t)− xf (�, t)|| �∫ t

0||�(�s, t − s)(fnk (s)− f (s))|| ds

� Mte t∫ t

0||fnk (s)− f (s)|| ds

� Mte t�E(t)||fnk − f ||E(X)for all k ∈ N, � ∈ �, t�0, which implies that

(Vfnk )(t)(Cb(�,X))−→ (Vf )(t) for all t�0.

It follows, using again the Remark 2.2, thatVf = g, and henceV is closed and so,by the closed-graph theorem it is also bounded. So we obtain that

||uf ||F(Cb(�,X)) = ||Vf ||F(Cb(�,X))� ||V || ||f ||E(X) for all f ∈ E(X) as required.

�

Lemma 3.2. If F is a Schäffer space, h ∈ F , h�0 and there exist two constantsa, b > 0 such thath(r)�ah(t)+ b, for all r� t�0 with r − t�1 then h ∈ L∞.

Proof. By the hypothesis we have that

h(n+ 1)�ah(s)+ b for all n ∈ N and alls ∈ [n, n+ 1]

and from here

h(n+ 1)�a∫ n+1

n

h(s) ds + b�a�F (1)||h||F + b for all n ∈ N

which implies that

c = supn∈N

h(n) < ∞.

Using again the hypothesis, we obtain that

h(t)�ah(n)+ b�ac + b for all n ∈ N and allt ∈ [n, n+ 1]. �

We consider againE andF two Schäffer spaces and we have:


Lemma 3.3. If the pair (E, F ) is admissible to�, then the following statements hold:(i) For all f ∈ E(X) there exista, b > 0 such that

|||uf (r)|||�a|||uf (t)||| + b f or all r� t�0 with r − t�1;

(ii) the pair (E,L∞) is admissible to�.

Proof. (i) We have that

xf (�, r) =∫ r

0�(�s, r − s)f (s) ds

=∫ t

0�((�s)(t − s), r − t)�(�s, t − s)f (s) ds +

∫ r

t

�(�s, r − s)f (s) ds

= �(�t, r − t)xf (�, t)+∫ r

t

�(�s, r − s)f (s) ds for all � ∈ �, r� t�0.

It results that

||xf (�, r)|| � Me (r−t)|||uf (t)||| +∫ r

t

Me (r−s)||f (s)|| ds

� Me |||uf (t)||| +Me ∫ t+1

t

||f (s)|| ds

� Me |||uf (t)||| +Me �E(1)||f ||E(X)

for all � ∈ �, r� t�0 with r − t�1. �

The condition (ii) follows directly from (i) and Lemma 3.2.

Lemma 3.4. If h1, h2 : R+ → R+ satisfy the following conditions:(i) h1(t)�h1(s)h2(t − s) for all t�s�0;(ii) sup

t∈[0,a]h2(s) < ∞, for all a > 0;

(iii) inft�0

h2(t) < 1,

then there exist two constantsN, � > 0 such that

h1(t)�Ne−�(t−s)h1(s) for all t�s�0.

Proof. By (iii) there exist > 0, � ∈ (0,1) such thath2() < �. Let t�s�0and n = [ t−s ], the largest integer less or equal thant−s . Then we have


that

h1(t) � h2(t − s − n)h1(n + s)� supv∈[0,]

h2(v)h1(n + s)

� supv∈[0,]

h2(v)hn1()h1(s)� sup

v∈[0,]h2(v)�

t−s −1h1(s)

= Ne−�(t−s)h1(s), where N =supv∈[0,]

h2(v)

�, � = − ln �

,

as required. �

Theorem 3.1. � is u.e.s if and only if there exists a pair(E, F ) of Schäffer spaces,admissible to�, with lim t→∞ �E(t)�F (t) = ∞.

Proof. Necessity: It follows easily from Definition 2.4 that the pair(L∞, L∞) isadmissible to�.Sufficiency: First observe that if the pair(E, F ) is admissible to�, then by Lemma

3.3 the pair(E,L∞) is admissible to�.Let x ∈ X, �0 ∈ � and f : R+ → X,

f (t) ={

�(�0, t)x, t ∈ [0,1],0, t > 1.

It is easy to check thatf ∈ E(X) and ||f ||E(X)�Me �E(1) ||x|| and

xf (�0, t) =∫ 1

0�(�0s, t − s)�(�0, s)x ds = �(�0, t)x

for all t�1 which implies that

||�(�0, t)x|| = ||xf (�0, t)||� |||uf (t)|||� ||uf ||L∞(Cb(�,X))�K||f ||E(X)�KMe �E(1)||x||

for all t�1, �0 ∈ � and all x ∈ X.Hence there existsL = Me max{K�E(1),1} such that

||�(�0, t)||�L and for allt�0, � ∈ �.


Let us consider again�0 ∈ �, > 0, x ∈ X and g : R+ → X

g(t) ={

�(�0, t)x, t ∈ [0, ],0, t�.

Then g ∈ E(X), ||g||E(X)�L�E()||x||.It follows that

xg(�0, t) =∫ t

0�(�0s, t − s)g(s) ds =

∫ t

0�(�0s, t − s)�(�0, s)x ds = t�(�0, t)x

for all t ∈ [0, ] and so

2

2||�(�0, )x|| =

∫

0s||�(�0, )x|| ds�

∫

0s||�(�0s, − s)�(�0, s)x|| ds

� L

∫

0||xg(�0, s)|| ds�L

∫

0|||ug(s)||| ds�L�F ()||ug||F(Cb(�,X))

� KL�F ()||g||E(X)�KL2�F ()�E()||x||.

Using (∗∗) we obtain that

�E()�F ()||�(�, t)||�8KL2 for all � ∈ �, > 0.

Applying Lemma 3.4 to the functionsh1, h2 : R+ → R+ defined by

h1(t) = sup�∈�

||�(�, t)||, h2(t) = 8KL2 + L

�E(t)�F (t)+ 1

it results that� is u.e.s. �

As an application of Theorem 3.1 we present the example which shows in somemanner the importance of our result.

Example 3.1. Let X = L1, � = R2+, T = {T (t)}t�0 the C0-semigroup onX givenby

(T (t)f )(s) = e−t f (t + s)


with the generatorA. Also we consider� : � × R+ → �, B : � → B(X) themappings defined by

�(�1, �2, t) = (�1 + t, �2e−t ),

(B(�1, �2)f )(s) ={e−�21(a(�2)+ 1− s)f (s), s�a(�2),e−�21f (s), s > a(�2),

where a(u) = min{1, u}. It is easy to check that� is a semiflow on� and B is astrongly continuous and bounded map on�. As it was shown in[1, Theorem 5.1]thedifferential equation

x′(t) = (A+ B(�t))x(t),

generates a strongly continuous skew-product semiflow� = (�,�) which is not nec-essary a solution in the classical sense but is a solution in themild sense i.e.:

�(�, t)x = T (t)x +∫ t

0T (t − s)B(�s)�(�, s)x ds, � ∈ �, t�0, x ∈ X.

If we takeE = F = L∞, f ∈ E, then

‖xf (�, t)‖�∫ t

0‖�(��, t − �)f (�)‖ d��

∫ t

0Me (t−�)‖f ‖E d� = M

e t‖f ‖E

for all � ∈ �, t�0. On the other hand, we have that

xf (�, t) =∫ t

0�(��, t − �)f (�) d�

=∫ t

0

(T (t − �)f (�)+

∫ t−�

0T (t − � − s)B((��)s)�(��, s)f (�) ds

)d�

=∫ t

0T (t − �)f (�) d� +

∫ t

0

∫ t−�

0T (t − � − s)B(�(s + �))�(��, s)f (�) ds d�

=∫ t

0T (t − �)f (�) d� +

∫ t

0

∫ t

�T (t − �)B(��)�(��, � − �)f (�) d� d�

=∫ t

0T (t − �)f (�) d� +

∫ t

0

∫ �

0T (t − �)B(��)�(��, � − �)f (�) d� d�,

=∫ t

0T (t − �)f (�) d� +

∫ t

0T (t − �)B(��)xf (�, �) d�


for all (�, t) ∈ � × R+, which implies that

‖xf (�, t)‖ �∫ t

0e−(t−�)‖f (�)‖ d� +

∫ t

0‖B(��)‖‖xf (�, �)‖ d�

� (1− e−t )‖f ‖E +∫ t

02e−(�1+�)2+ � M

‖f ‖E d�

�(1+ 2M

∫ ∞

0e−�2+ � d�

)‖f ‖E

for all t�0. Henceuf ∈ F(Cb(�, X)). It follows that � is uniformly exponentiallystable.We see that to verify that some pair(E, F ) is admissible to� is in fact to check

that a function is inF (not necessarily exponentially bounded) with the additionalhypothesis that some function belongs toE.

Now we conclude with

Theorem 3.2. The following assertions are equivalent

(1) � is u.e.s;(2) there existsE a Schäffer space such that the pair(E,E) is admissible to�;(3) there existp, q ∈ [1,∞], (p, q) �= (1,∞) such that the pair(Lp, Lq) is a

admissible to�;(4) there existp, q ∈ [1,∞) such that the pair(Mp,Mq) is admissible to�;(5) there existp ∈ (1,∞], q ∈ [1,∞) such that the pair(Lp,Mq) is admissible

to �.

Proof. Follows easily from Theorem 3.1 and Example 2.2.�

Remark 2.4. From the statement (2) of the Theorem 3.2 and Example 2.1 it followsalso that� is u.e.s. if and only if there exists an Orlicz spaceL�, such that the pair(L�, L�) is admissible to�.

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Schäffer spaces and uniform exponential stability of linear skew-product semiflows